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# Modular reduction in Galois fields

I want to compute x^6 mod x^5+x^2+1 in the Galois Field GF(2^5). Does anyone know how to do this in SAGE?

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First, define k to be the field GF(2^5), whose generator is named a:

sage: k.<a> = FiniteField(2^5); k
Finite Field in a of size 2^5


Alternatively :

sage: k = GF(2^5, 'a'); k
Finite Field in a of size 2^5


Then, define the polynomial ring k[x]:

sage: R.<x> = PolynomialRing(k); R
Univariate Polynomial Ring in x over Finite Field in a of size 2^5


Alternatively:

sage: R = k['x']; R
Univariate Polynomial Ring in x over Finite Field in a of size 2^5


Then, do your computation in R:

sage: P = R(x^6)
sage: P.mod(x^5+x^2+1)
x^3 + x

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## Comments

Thank you very much - helped me a lot!

( 2013-04-28 01:58:42 -0600 )edit

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Asked: 2013-04-28 00:04:04 -0600

Seen: 665 times

Last updated: Apr 28 '13